In information theory, a main application of the EPI is to derive extremal
entropy inequalities, which can then be used to solve network communication
problems. In their work [1]}, Liu and Viswanath derived an extremal
entropy inequality based on the classical EPI of Shannon  and
Stam  and used it to establish the private message capacity
region of the vector Gaussian broadcast channel via the Marton outer bound
[2]}. In this paper, we will derive a new extremal
entropy inequality based on the generalized Costa EPI and use it to
characterize the secrecy capacity regions of the degraded vector Gaussian
broadcast channel with layered confidential messages.
